As the radius ratio increases, what we see is that there is a corresponding

change in the coordination number at certain specific radius ratios.

Once we have identified the behavior of the coordination number, we can see how it

leads directly into packing to form crystal structures in these ionic salts.

The coordination number and

the radius ratio go hand in hand with respect to one another.

When we look at a coordination number of 2, for example, which is the simplest of

all coordination numbers, what we have is on either side of the cation, the anion.

And if we consider this like a daisy chain or a string of pearls,

we see that we will have the alternating anion, cation, anion, cation.

And what this winds up doing to the material is that it will establish

an overall electrostatic balance on the short range as well as the long range

with respect to the positively charged cation and the negatively charged anion.

If we consider higher coordination numbers.

For example if we draw a figure of three circles and we position those so

that they are at the centers of the vertices of an equilateral triangle and

we look at the figure on the right hand side, we have developed a critical

radius ratio that for a cation and an anion,

we have the anions just touching one another as you can see on

the picture to the right and the cations are just touching the anions.

And if we construct a triangle which is given there, which is a right triangle,

and we know the relationship that exists between a 30, 60,

90 degree triangle, namely that of 1, 2, square root of 3.

We can then determine what that radius ratio is and

that critical radius ratio turns out to be a value of 0.155.

And it's determined by using that triangular relationship.

So the dimension that is on the side which measures the cation,

anion distance, that's the hypotenuse of the right triangle.

And so what we have then is the fact that we have little r plus big r for

the hypotenuse.

When we look at the one side, the base of that right triangle,

the big R represents then the radius.

So we now can use those relationships to describe that critical value of 0.155.

Now, if we were to take and go below that critical value